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In this paper, we first prove a uniform upper bound on costs of null controlsfor semilinear heat equations with globally Lipschitz nonlinearity on asequence of increasing domains, where the controls are acted on anequidistributed set that spreads out in the whole Euclidean space R^N . As anapplication, we then show the exactly null controllability for this semilinearheat equation in R^N . The main novelty here is that the upper bound on costsof null controls for such kind of equations in large but bounded domains can bemade uniformly with respect to the sizes of domains under consideration. Thelatter is crucial when one uses a suitable approximation argument to derive theglobal null controllability for the semilinear heat equation in R^N . Thisallows us to overcome the well-known problem of the lack of compactnessembedding arising in the study of null controllability for nonlinear PDEs ingenerally unbounded domains.

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application